Really tiny

M.C.'s mysterious mentor has some strange ideas regarding calculus:

...Objections like those raised by Berkeley inspired the epsilon-delta definition of a limit and... the eventual formal arithmetization of calculus. But, from what I understand, these mathematicians, though they all made important contributions to the development of modern calculus, were not interested in preserving the notion of infinitesimal value. In point of fact, as most histories of mathematics will clearly attest, after the work of Weierstrass, the notion of infinitesimal value had been altogether purged from the mathematical literature, and epsilon-delta limits were used instead to define integrals, derivatives, convergent sequences, and all similar notions... [emphasis mine]

Except that the entire driving purpose behind the epsilon-delta definition of the limit was to quantify the infinitesimal!

As the definition indicates, no matter how small the distance betweeen f(x) and the limit y_0 (which is the epsilon), you can find a delta-neighborhood around x_0 such that the epsilon-bound is satisfied. This is what makes writing a epsilon-delta proof so difficult -- you have to make sure that no matter how infinitesimally small (or infinitely large) epsilon gets, that your delta is the appropriate size to ensure the epsilon-bound holds.

If you don't believe me, then I'll have to quote the textbook "Elementary Real Analysis" by Thomson, Bruckner, and Bruckner:

The study of convergent sequences was undertaken and developed in the eighteenth century without any precise definition. The closest one might find to a definition in the early literature would have been something like

A sequence converges to a number L if the terms of the sequence get closer and closer to L.

Apart from being too vague to be used as anything but a rough guide for the intuition, this is misleading in other respects... We want not merely "closer and closer" but somehow a notion of "arbitrarily close".
The definition that captured the idea in the best way was given by Augustin Cauchy in the 1820s. He found a formulation that expressed the idea of "arbitrarily close" using inequalities. In this way the notion of limit is defined by a straightforward mathematical statement about inequalities. (p. 33)

"Arbitrarily close". This is what makes Cauchy's idea so revolutionary -- he did not need to define "infinitesimal" in order to deal with the infinitesimal, because epsilon could be as small as you wanted! If you want to let epsilon equal 1, you can find an appropriate delta-neighborhood. If you want to let epsilon equal 0.0000000000001, you still can find an appropriate delta-neighborhood. Thus mathematicians were finally able to precisely explain what they meant when a quantity approached another quantity, i.e. Newton's fluxions. (Mathematicians were also able to torture their analysis students by making them construct countless epsilon-delta proofs.)

Cauchy's developments would seem to contradict M.C. and the Mentor's claim that Robinson was the first to give the notion of "infinitesimal" sound mathematical footing.